We define the operator: $l^p$ to $l^p$ with the $p$-norm
$T( (a_{n} ))= a_{n+1}-a_{n}$ So its like:
$(a_1,a_2,a_3.....)$ goes to $(a_2-a_1,a_3-a_2....)$
They re asking me about the norm of the operator, first I see that $|| (1,0,0,0...) ||_p=1$ and $|| T((1,0,0..)=1 ||_p$ so the norm is more or equal than 1. But for proving that the norm is less or equal than 1 I have a problem, because $|| a_{n+1}-a_{n} ||_p$ I can see for triangle inequality that is less or equal than 2 but not that 1. Thank you
Define $a_n:=c^n$ where $\left|c\right|\lt 1$. Then $\left|a_{n+1}-a_n\right|^p=c^{np}\left|c-1\right|^p=a_n^p\left|c-1\right|^p$, which gives that $\left\lVert T\left(\left(a_n\right)_{n\geqslant 1}\right)\right\rVert =\left|c-1\right|\left\lVert\left(a_n\right)_{n\geqslant 1}\right\rVert $. We thus obtain that $\lVert T\rVert\geqslant\left|c-1\right| $ for each $c\in (-1,1)$. Now let $c\to -1$ to get the result.